integral of x(x+7)^{-3/2}

Lösung

\int x (x + 7)^{- \frac{3}{2}} d x

2 (x + 7)^{\frac{1}{2}} + \frac{14}{( x + 7 ) ^{\frac{1}{2}}} + C

Schritte zur Lösung

\int x (x + 7)^{- \frac{3}{2}} d x

Vereinfache

x (x + 7)^{- \frac{3}{2}} : \frac{x}{( x + 7 ) ^{\frac{3}{2}}}

= \int \frac{x}{( x + 7 ) ^{\frac{3}{2}}} d x

Wende U-Substitution an

= \int \frac{u - 7}{u ^{\frac{3}{2}}} d u

Multipliziere aus

\frac{u - 7}{u ^{\frac{3}{2}}} : \frac{1}{u ^{\frac{1}{2}}} - \frac{7}{u ^{\frac{3}{2}}}

= \int \frac{1}{u ^{\frac{1}{2}}} - \frac{7}{u ^{\frac{3}{2}}} d u

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{1}{u ^{\frac{1}{2}}} d u - \int \frac{7}{u ^{\frac{3}{2}}} d u

\int \frac{1}{u ^{\frac{1}{2}}} d u = 2 u^{\frac{1}{2}}

\int \frac{7}{u ^{\frac{3}{2}}} d u = - \frac{14}{u ^{\frac{1}{2}}}

= 2 u^{\frac{1}{2}} - (- \frac{14}{u ^{\frac{1}{2}}})

Setze in

u = x + 7

ein

= 2 (x + 7)^{\frac{1}{2}} - (- \frac{14}{( x + 7 ) ^{\frac{1}{2}}})

Vereinfache

= 2 (x + 7)^{\frac{1}{2}} + \frac{14}{( x + 7 ) ^{\frac{1}{2}}}

Füge eine Konstante zur Lösung hinzu

= 2 (x + 7)^{\frac{1}{2}} + \frac{14}{( x + 7 ) ^{\frac{1}{2}}} + C

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